Using standard tools of harmonic analysis, we state and solve the problem
of moments for non-negative measures supported on the unit ball of a Sobolev
space of multivariate periodic trigonometric functions. We describe outer and
inner semidefinite approximations of the cone of Sobolev moments. They
are the basic components of an infinite-dimensional moment-sums of squares
hierarchy, allowing to numerically solve non-convex polynomial optimization
problems on infinite-dimensional Sobolev spaces with global convergence guarantees.
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